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How to concentrate idempotents. (English) Zbl 1202.42006

An idempotent is defined as a finite sum \[ f(x)= \exp(2\pi N_1x)+\cdots\exp(2\pi N_mx), \] where \(N_1,\dots, N_m\) are integers. It is said that there holds \(L^p\)-concentration, if there exists a \(c_p> 0\) such that for each interval \(I\subset\mathbb{T}= [-1/2,1/2]\) and each \(\varepsilon> 0\) there exists an idempotent \(f\) such that \[ \Biggl(\int_I |f(x)|^p dx\Biggr)/ \Biggl(\int_{\mathbb{T}} |f(x)|^p dx\Biggr)> c_p- \varepsilon. \] The paper is a survey of results concerning existence of \(L^p\)-concentration for \(p> 0\).

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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